A, B, C are three non-null square matrices of the same order, wri… - Vidyadip

Question

$A, B, C$ are three non-null square matrices of the same order, write the condition on $A$ such that $AB = AC \Rightarrow B = C.$

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Answer

Consider $AB = AC.$
On multiplying both sides by $A^{-1},$
we get $AA^{-1}B = AA^{-1}$
$ \Rightarrow IB = IC [$Because $AA^{-1} = I$ where I is the identity matrix$] $
$\Rightarrow B = C$
Therefore, the required condition is $A $ must be invertible or $|\text{A}|\neq0$.

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